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%% section 3 Relaxation of the Phases [slacpub7010003 in slacpub7010003: slacpub7010004]
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\section{\usemenu{slacpub7010::context::slacpub7010003}{Relaxation of the Phases}}\label{section::slacpub7010003}
In order to motivate the relaxation solution to the SUSY CP
problem it is instructive to review the role of
nonlinearly realized global symmetries in this context.
The nonsupersymmetric standard model has, at the renormalizable
level, two accidental global symmetries, namely baryon and
lepton number.
If these symmetries are realized nonlinearly there are Goldstone
bosons which couple to the associated currents.
However, if the scale of spontaneous symmetry breaking is
large enough, the Goldstone bosons decouple and there
are no effects at low energy \cite{16}.
The two Higgs doublet model has, in the absence of an
$m_{12}^2 H_u H_d$ term, an additional global PecceiQuinn
symmetry at the classical level \cite{11}.
This symmetry has a quantum mechanical anomaly with respect
to QCD, so $\bar{\theta}_{QCD}$ shifts under a PecceiQuinn transformation.
If this symmetry is realized in the Goldstone mode, i.e.
Arg$(m_{12}^2)$ is a dynamical variable, the associated
pseudoGoldstone boson (the axion) receives a potential from
the explicit breaking due to the anomaly.
It is technically natural that this potential is an extremum at
points of enhanced symmetry.
This is because if the symmetry is realized (nonlinearly or otherwise)
in the relevant degrees of freedom, then the lowest order term in
the potential near a symmetry point is bilinear in the fields.
If such points are in fact minima, then
since $\bar{\theta}_{QCD} \rightarrow  \bar{\theta}_{QCD}$ under CP,
the axion can relax to
a CP conserving point, $\bar{\theta}_{QCD} = 0$ or $\pi$.
The explicit breaking
from the anomaly comes from a marginal operator, the topological
charge density.
Since QCD is asymptotically free, the low energy long distance
dynamics {\it can} in principle
determine the potential for the axion (as is usually implicitly
assumed).
Here low energy refers to the standard model particle content with
renormalizable interactions.
It is important to note that there can be additional
explicit breakings from high energy short distance
physics, which may disturb the alignment.
High energy refers to for example GUT or
Planck scale physics, which may contain additional degrees of
freedom which do not
conserve CP with respect to the low energy theory.
However, in order for the mechanism to work, one must assume the
PecceiQuinn symmetry is respected by the high energy physics to
a sufficiently high order in irrelevant operators \cite{17}.
%comment symmetyr in hidden sector  coupling to visible
%sector explicitly breaks 
%phases turn out to correspond to the CP vilating phases.
% potential from low energy dynamamics minima at (or near)
%CP conserving points.
The solution of the SUSYCP problem we propose is to
promote the phases appearing in (\docLink{slacpub7010002.tcx}[muterm]{1}) and
(\docLink{slacpub7010002.tcx}[softterms]{2}) to dynamical variables.
The soft terms of course arise from couplings with the
SUSY breaking sector.
Since the $\mu$ term must be of order the weak scale, the
only reasonable assumption is that it too arises from
a coupling to the SUSY breaking sector.
Promoting the phases to dynamical variables therefore amounts
to postulating the existence of spontaneously broken
global symmetries in the SUSY breaking sector.
%Spontaneous broken global symmetries are common in
%models of dynamical SUSY breaking.
%In fact, all know models of SUSY breaking which rely
%on a dynamically generated superpotential contain
%an accidental $R$ symmetry which is spontaneously broken.
The phases in ({\docLink{slacpub7010002.tcx}[muterm]{1}) and (\docLink{slacpub7010002.tcx}[softterms]{2}) are then
the Goldstone bosons of these nonlinearly realized symmetries.
In general all the phases need not be dynamical.
However, for now we make the ``maximal'' assumption that
all four phases are dynamical and investigate the consequences.
As we show below, one phase is analogous to baryon and lepton
number in that it is respected at the renormalizable level by
couplings to the visible sector.
The Goldstone boson which could be associated with this symmetry
therefore has no effect at low energy if the SUSY breaking scale
is large enough.
A second symmetry is analogous to the PecceiQuinn symmetry
in the two Higgs doublet model, and can lead to the
PecceiQuinn mechanism for the relaxation of
$\bar{\theta}_{QCD}$ \cite{18}
(if respected to high enough order).
The remaining two phases are the physical SUSY CP violating phases.
As we show below
any symmetries associated with these turn out
to be explicitly broken by couplings with the visible sector.
Integrating out the visible sector then produces a potential with
minima near CP conserving points for the phases.
This leads to a relaxation mechanism analogous to the
PecceiQuinn mechanism.
%put a note somewhere about which phases get a potential and which
%dont . the ones that correspond to the physical CP violating
%phases have a potential cos.,, therefore relax to CP conserving
%point.
The relaxation mechanism requires that %the Goldstone bosons
%associated to the phases (\ref{combinations}) are lifted and that
the vacuum energy
be a minimum at (or near) a CP conserving point
for the phases (\docLink{slacpub7010002.tcx}[combinations]{3}).
The global symmetries associated with the phases
must therefore be explicitly broken by some couplings.
The possible terms appearing in the vacuum energy
which depend on the phases are limited by the selection
rules given above.
Treating the dimensionful parameters as background spurions
and including the Higgs bosons, which acquire an expectation
value at low energy, the possible
phase dependent terms are
\beq
\begin{array}{cc}
m_{\lambda} \mu H_u H_d & m_{\lambda} \mu (m_{12}^2)^* \\
& \\
A \mu H_u H_d & A \mu (m_{12}^2)^* \\
& \\
A^* m_{\lambda} H_u^* H_u & A^* m_{\lambda} H_d^* H_d \\
& \\
A^* m_{\lambda} \Lambda^2 & \\
\end{array}
\label{vacterms}
\eq
where each term appears with $+~h.c.$, and
$\Lambda$ is a dimensionful scale discussed below.
In general there are contributions to the terms above
from both long and short distance physics.
The long distance contributions from light degrees of freedom come from
diagrams such as those given in Figs. 14.
%First consider the contribution from integrating out low energy
%degrees of freedom.
%Examples of such vacuum diagrams are given in Figs. 14.
Just on dimensional grounds all the terms
in (\docLink{slacpub7010003.tcx}[vacterms]{4}), except the last one,
correspond to marginal operators.
For the marginal operators, at worst, all
logarithmic energy scales contribute equally to the coefficients.
The contributions from integrating out light degrees of freedom can
therefore dominate the short distance contributions by
${\cal O}(\ln(\Lambda^2/m_W^2))$, where $m_W$ is the weak scale
and $\Lambda$ is the scale
at which SUSY breaking is transmitted to the visible
sector ($\Lambda \sim M_p$ in hidden sector models).
In this case the breaking of the global symmetries can come
mainly from the %coupling of the SUSY breaking sector to the
visible sector, and need not be
particularly sensitive to short distance physics.
This relative insensitivity to short distance physics
is in contrast to the situation for dynamical squark flavor
matrices \cite{10}, dynamical Yukawa
couplings \cite{19,20}, or a dynamical determination of
the SUSY breaking scale in noscale type models \cite{21,22};
in these cases the potential is
quadratically sensitive to the short distance physics \cite{23}.
Likewise here, the last term in (\docLink{slacpub7010003.tcx}[vacterms]{4}) corresponds to
a relevant operator.
Integrating out light degrees of freedom therefore gives
(in the absence of a regulator) a
quadratically divergent contribution to the operator,
proportional to $\Lambda^2$.
An example of such a three loop diagram is obtained from Fig. 3.
with $H_u$ contracted with $H_u^*$.
Because of the quadratic divergence, this contribution to the
vacuum energy is very sensitive to the short distance physics.
This sensitivity implies
the precise description of the short distance contribution is
in fact scheme dependent.
For example, in dimensional regularization there are no quadratic
divergences, and the $\Lambda^2$ piece comes from the matching
conditions at the scale $\Lambda$.
The potential for the phase Arg$(A^* m_{\lambda})$ is
therefore essentially determined by physics at the scale $\Lambda$.
Now consider the form of the potential
arising from (\docLink{slacpub7010003.tcx}[vacterms]{4}) for the
Goldstone bosons associated to the CP violating phases
in (\docLink{slacpub7010002.tcx}[combinations]{3}).
First note that in the ground state the $m_{12}^2 H_u H_d$
term in the potential fixes
Arg$(H_u H_d)$ = $$Arg$(m_{12}^2)$.
Ignoring for the moment the KobayashiMaskawa phase and
any flavor changing phases,
the long distance contributions
of the types given in Figs. 14
to the marginal operators then all go like
\beq
\sum_i
c_i m_W^4 ~\cos(\phi_{\alpha} + \delta_i)
\label{sumlong}
\eq
where $c_i$ is the magnitude of the $i$th diagram,
%$\delta_i = 0$ or $\pi$,
and $\phi_{\alpha}$ = Arg$(m_{\lambda} \mu (m_{12}^2)^*)$ or
Arg$(A \mu (m_{12}^2)^*)$.
Since the only CP violating phase in the lowest order diagrams
is the phase $\phi_{\alpha}$ itself, $\delta_i = 0$ or $\pi$.
The lowest order long distance contributions to the vacuum
energy therefore
necessarily have minima at CP conserving
points.
%Some of the extrema can be minima, which is in fact likely
%as a single diagram typically gives the dominant
%long distance contribution, and therefore dominates the
%sum in (\ref{sumlong}).
%(although in general there may also be minima at other points).
%For example, diagrams involving Yukawa couplings, such
%as those in Figs. 2 and 3, are dominated by the top.
%As argued above, for the marginal operators the long distance
%can dominate the short distance contributions.
For the relevant operator the lowest order contributions go like
\beq
\sum_i
c_i m_W^2 \Lambda^2 ~\cos(\phi +\delta_i)
\eq
where $\phi$ = Arg$(A^* m_{\lambda})$.
Again, the long distance contributions, such as that
in Fig. 3, give $\delta_i = 0$ or $\pi$.
The short distance pieces, however, in general have arbitrary
$\delta$.
In order to proceed without simply assuming a tuning of the short
distance phases
we must therefore assume that the global
symmetry associated with Arg$(A^* m_{\lambda})$ is realized in
the short distance physics at the scale $\Lambda$.
In addition we must assume that the
short distance physics has
the same definition of CP as the long distance physics, so that the
potential has extrema at
the CP conserving points,
$\delta = 0$ or $\pi$ \cite{24}.
%As this is a point of enhanced symmetry it is of course technically
%natural to postulate that the potential
%is a minimum there \cite{relaxnote}.
This could occur for example if CP is a symmetry of the full theory,
and only broken spontaneously below the scale $\Lambda$.
In string theory, where CP is a symmetry \cite{25},
with hidden sector SUSY breaking this could occur if the
scale of spontaneous CP violation
is between the Planck and intermediate
SUSY breaking scale $M_I \sim \sqrt{m_W M_p}$.
Note that without the relaxation mechanism the soft parameters and
$\mu$ would not in general be real in this case.
With these assumptions about the short distance physics,
since the combinations of phases that
appear in the vacuum energy are precisely those that appear
in any CP odd observable, there is a ground state in which all
physical amplitudes (proportional to the terms in (\docLink{slacpub7010002.tcx}[combinations]{3}))
are CP conserving.
This can also be seen by
starting from the original basis for the phases in (\docLink{slacpub7010002.tcx}[muterm]{1}) and
(\docLink{slacpub7010002.tcx}[softterms]{2}).
$U(1)_{PQ}$ and $U(1)_{RPQ}$ redefinitions may always
be used to transform to a basis in which any two of the phases vanish,
for example
$\phi_A = \phi_{m_{12}^2} =0$.
In the CP conserving ground state the alignment of
Arg$(A^* m_{\lambda})$ then forces $\phi_{m_{\lambda}}=0$ or $\pi$,
and the alignment of Arg$(A \mu (m_{12}^2)^*)$ forces
$\phi_{\mu} = 0$ or $\pi$.
The alignment of the phases described above can be disturbed
in a number of ways.
Higher order loop diagrams of light degrees of freedom
can be proportional to products of the invariants (\docLink{slacpub7010003.tcx}[vacterms]{4}),
and therefore have potentials proportional
to $\cos( n \phi_{\alpha} \pm n^{\prime} \phi_{\beta} + \delta )$.
These however are suppressed by at least two additional loop factors
%${\cal O}(\alpha / 4 \pi)^2$,
and do not shift the CP conserving minima.
The KobayashiMaskawa phase can in principle shift the minimum
of the potential from a CP conserving point.
To form the Jarlskog invariant
%$J= {\rm Im} (V_{us}V_{cs}^*V_{cd}V_{ud}^*)$
$J= {\rm Im} (V_{ud}V_{td}^*V_{tb}V_{ub}^*)$
however requires
at least four $SU(2)$ gauge couplings.
This requires at least two additional loops compared with the
lowest order diagrams, and is therefore down by
at least ${\cal O}((\alpha / 4 \pi)^2 J)$.
GIM suppression among the squarks would reduce this contribution
even further.
The KobayashiMaskawa phase therefore does not significantly shift
the minima from CP conserving points.
The alignment can also be disturbed by
explicit breaking by the short distance physics of
both CP symmetry and
the
global symmetries in the hidden
sector which make the phases dynamical.
Here, just as for the PecceiQuinn mechanism, we must assume that
the minimum is not shifted to high enough order in irrelevant operators.
However, since the bound on the phases is
${\cal O}(10^{2}10^{3})$ this is not nearly as restrictive as
for the axion.
%since the phases only need to be smaller than
%${\cal O}(10^{2}10^{3})$ this is not very restrictive.
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